sample_tt1_1_ans

sample_tt1_1_ans - Math 235 Sample Term Test 1 - 1 Answers...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 235 Sample Term Test 1 - 1 Answers NOTE : - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Give the definition of an inner product h , i on a vector space V . Solution: h , i : V × V → R such that h ~v,~v i > 0 for all ~v 6 = ~ 0 and h ~ , ~ i = 0. h ~v, ~w i = h ~w,~v i h ~v,a~w + b~x i = a h ~v, ~w i + b h ~v,~x i b) Let B = { ~v 1 ,...,~v n } be orthonormal in an inner product space V and let ~v ∈ V such that ~v = a 1 ~v 1 + ··· + a n ~v n . Prove that a i = < ~v,~v i > . Solution: Taking the inner product of both sides with ~v i to get < ~v,~v i > = < a 1 ~v 1 + ··· + a n ~v n ,~v i > = a 1 < ~v 1 ,~v i > + ··· + a n < ~v n ,~v i > = a i since B is orthonormal. c) Define what it means for a set B to be orthonormal in an inner product space V . Solution: A set B = { ~v 1 ,...,~v n } is orthonormal in V if < ~v i ,~v j > = 0 for all i 6 = j and < ~v i ,~v i > = 1 for 1 ≤ i ≤ n ....
View Full Document

This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

Page1 / 3

sample_tt1_1_ans - Math 235 Sample Term Test 1 - 1 Answers...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online